First part $(\implies)$ Because $x_n$ are non-negative terms then the series $\sum x_n$ and $\sum \frac{x_n}{1+x_n}$ increases. If $\sum x_n$ does not diverge to infinity then is bounded above, and then converges because the series is increasing.

Then I want to show that if $\sum x_n$ converges then $\sum\frac{x_n}{1+x_n}$ converges too.

Suppose that $\sum x_n\to s$. It is enough to observe that if $x_k\ge 0$ then $\frac{x_k}{1+x_k}\le x_k$ for any $k$ then

$$\sum\frac{x_n}{1+x_n}\le\sum x_n\to s$$

Because $\sum\frac{x_n}{1+x_n}$ is bounded above by $s$ and is increasing then converges.$\Box$